Risk Latte - The Simple Mathematics of Portfolio Insurance

The Simple Mathematics of Portfolio Insurance

Team Latte
November 22, 2011

With global equity markets gyrating wildly, many institutional investors are wondering if after all the time has come to take the Portfolio Insurance strategy out of the investment closet once again. Much has been said about portfolio insurance and its close cousin Constant Proportion Portfolio Insurance (CPPI) in the financial press as well as in serious investment literature over the decades. Many strategists and market observers blame portfolio insurance for market crashes and believe that this investment strategy, designed to protect against market crashes, actually intensifies the downward spiral of the market more.

So, what is portfolio insurance? How can we explain this concept in the simplest possible terms?

A portfolio insurance strategy is simply the asset plus a put option on the asset. In other words, if an investor buys an asset, say a stock or a stock index, and also buys a put option on that asset then he creates a portfolio insurance strategy. The put option creates a floor on the investment losses. Say, you have $100 to invest and you donít want to lose more than $10, i.e. your portfolio should not breach the $90 level at maturity (at a certain investment horizon). You then buy an asset for $100 and buy a put option with a strike price of 90 on the asset trading at 100. Of course, the arithmetic will not add up because the cost of buying the put is over and above the $100 that you have to invest, and thereís a way how one can get around this problem. But thatís not the issue here.

All we want to show that if we buy an asset for 100 and a put with a strike of 90 then we can create a floor on our investment. This statement may seem obvious or even trivial to many traders and investors, but it can actually be proven mathematically.

Say, we want to buy an asset, , whose value at maturity, (or at a certain investment horizon) is given by and we choose a put option on this asset with a strike price of . Note, that the strike price, is constant. Therefore, the payoff of the put option at maturity is given by:

Now, if we go long (buy) the asset and long (buy) the put option on the asset then the payoff of the portfolio at maturity (investment horizon) will be:

With a bit of algebraic manipulation we can show that the payoff of the portfolio at time, is given by:

Thus, the effect of adding a put option to the asset is equal to a portfolio whose payoff is the maximum of either the value of the asset at maturity or the constant strike price. So, if we invest in an asset at 100 and buy a put option with a strike price of 90, then the minimum value of the portfolio at maturity will be 90 and the maximum would be unlimited depending upon the upside of the asset. In fact, we can choose any strike price, depending on what floor we want to put to our investment.

Of course, as mentioned above, the arithmetic of the above transaction is slightly more complicated due to transaction costs and the cost of the put option but getting around these issues is rather easy.

However, if too many investors start to implement such a strategy and start buying put options at a given strike (or at various strike prices) then such an action would put a downward pressure on the asset. Moreover, if the demand for put options starts to rise due to investorsí buying them to protect their asset positions, the volatility will start to go up as traders selling these puts will start to mark up the out of the money volatility causing more pressure on the asset via the increased volatility.

Also, note that investors donít actually have to buy put options. Many mutual fund managers are prohibited from trading derivatives. An investor can synthetically create a put and replicate the above strategy Ė and in fact, many are doing it, even as we write this article Ė to put floor on their investment. The effect of that on the market is the same. It creates bigger downward pressure on the asset.

Reference:  For more on Portfolio Insurance, read Mark P. Kritzmanís Asset Allocation for Institutional Investors (Richard D. Irwin, 1990).

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